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TrigonometricIdentity

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Description

Summary

Decides whether a rational function of sines, cosines, tangents, cotangents, secants and
cosecants is a constant where it has a value, and which constant. Proving a trigonometric
identity written as a difference is asking whether that constant is zero.

Remarks

Every angle has to be a combination of the variables with rational coefficients and no
constant term. Each variable v then has a base angle u = v / L, where L is the least common denominator of its coefficients, and every angle is a whole combination
of base angles. A cosine of n u is the Chebyshev polynomial T_n(cos u) and a sine
is sin(u) U_(n-1)(cos u), and a sum of base angles is opened by the addition theorem.
So the expression is a quotient of two polynomials over Q in its variables and in a
sine and a cosine of each base angle.
 The only relation among those is sin(u)^2 + cos(u)^2 = 1, since x and
e^(ix) are algebraically independent, and so are different variables. Writing each
sin(u)^2 as 1 - cos(u)^2 leaves every sine to the first power at most, and that
form is unique: the relations' leading monomials are coprime, so they are a Gröbner basis. A
polynomial therefore vanishes identically exactly when its form is zero, and the quotient is
the constant k exactly when the numerator's form is k times the denominator's.
 Both theorems hold for every complex angle, and the denominator is the product of the ones
the expression divides by, so it vanishes only where the expression has no value. The
constant is the expression's value everywhere else, and a caller states where that is.
#2142

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